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Abstract: Invariants at arbitrary and fixed energy strongly and weakly conservedquantities for 2-dimensional Hamiltonian systems are treated in a unified way.This is achieved by utilizing the Jacobi metric geometrization of the dynamics.Using Killing tensors we obtain an integrability condition for quadraticinvariants which involves an arbitrary analytic function $Sz$. For invariantsat arbitrary energy the function $Sz$ is a second degree polynomial with realsecond derivative. The integrability condition then reduces to Darboux-scondition for quadratic invariants at arbitrary energy. The four types ofclassical quadratic invariants for positive definite 2-dimensional Hamiltoniansare shown to correspond to certain conformal transformations. We derive theexplicit relation between invariants in the physical and Jacobi time gauges. Inthis way knowledge about the invariant in the physical time gauge enables oneto directly write down the components of the corresponding Killing tensor forthe Jacobi metric. We also discuss the possibility of searching for linear andquadratic invariants at fixed energy and its connection to the problem of thethird integral in galactic dynamics. In our approach linear and quadraticinvariants at fixed energy can be found by solving a linear ordinarydifferential equation of the first or second degree respectively.



Autor: Kjell Rosquist, Giuseppe Pucacco

Fuente: https://arxiv.org/







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